Decide if the following claims are true or false, providing either a short proof or counterexample to justify each conclusion. Assume throughout that g is defined and continuous on all of R.
(a) If $g(x)\geq 0$ for all $x<1$, then $g(1)\geq 0$ as well.
(b) If $g(r)=0$ for all $r \in Q$, then $g(x)=0$ for all $x \in R$.
(c) If $g(x_{0}) > 0$ for a single point $x_{0} \in R$, then $g(x)$ is in fact strictly positive for uncountably many points.
My solution:
Best Answer
(a) is nonsense, as $g$ is assumed to be continuous on all of $\mathbb{R}$, so in all points in particular. Suppose that $g(1) < 0$. Set $\varepsilon = \frac{-g(1)}{2} > 0$ and apply continuity on the left...
(b) is true, but your argument is insufficient here. Where do you use continuity and denseness? Do you apply to some theorem, and if so, which one?
(c) your example is false in two ways: it is not continuous (which it has to be), and it is strictly positive for all $x > 0$, so uncountably many $x$. Try to prove it instead. Same idea as (a): set $\varepsilon = \frac{g(x_0}{2} > 0$ and apply continuity at $x_0$.